Sharp Estimates of Pochhammer’s Products

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please see the PDF.

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper is interesting, however it lacks significantly due to the following reasons.

1. The paper does not reflect contemporary research in numerical methods developed for evaluation of special functions, especially Gamma function.

2. The paper does not compare the obtained results with the other estimates, in particular, thoese used to evaluate Gamma function.

3. The paper does not present any practical application of the obtained estimates.

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

1. Summary of the Manuscript

The manuscript presents sharp asymptotic estimates for classical and generalized rising and falling Pochhammer products with positive arguments. The author employs Stirling’s approximation formula for the Gamma function as the basis for deriving the results. The manuscript focuses exclusively on mathematical formulas without offering significant contextual discussion, comparison with existing literature, or practical implications.

2. Strengths and Weaknesses

Strengths:

• The topic is mathematically significant and contributes to the field of approximation theory.
• The derivation of estimates based on Stirling’s formula is a logical approach that aligns with the goals outlined in the abstract.
• The paper demonstrates technical proficiency in deriving mathematical expressions.

Weaknesses:

• Lack of Purpose: The paper does not explicitly define its purpose or articulate how these estimates contribute to the broader mathematical or applied context.
• Minimal Discussion: No meaningful comparison with existing literature or prior work is provided, which limits the reader's understanding of the novelty or relevance of the results.
• Organization: The manuscript’s structure is unconventional and does not follow standard conventions for scientific papers. Key sections, such as an introduction, methodology, results, and discussion, are either absent or not clearly delineated.
• Inconsistent Voice: The paper uses "we" and "our," implying multiple authorship, though only one author appears credited. This inconsistency diminishes clarity and professionalism.
• No Discussion of Error Terms: While the abstract mentions error terms, the paper does not adequately discuss their implications, significance, or magnitude.

3. Major Recommendations

1. Clearly Define the Purpose: Include a clear introduction that outlines the aim of the study, its significance, and how it builds on or diverges from previous works.
2. Organize the Paper Conventionally: Reorganize the manuscript into standard sections (e.g., Introduction, Methodology, Results, Discussion, and Conclusion). This structure will help readers follow the logical flow of the paper.
3. Compare with Literature: Provide a discussion that situates the results within the broader context of the field, highlighting advancements, limitations, or relationships to existing work.
4. Include Discussion and Implications: Beyond presenting formulas, discuss the practical implications, accuracy, and limitations of the derived estimates. If relevant, include potential applications.
5. Address Error Terms: Provide a detailed explanation of the error terms mentioned in the abstract, supported by examples or figures, to validate their accuracy and relevance.
6. Refine Abstract and Keywords: Revise the abstract to clarify the motivation and impact of the research. Avoid generic terms in keywords and focus on terms most relevant to the mathematical community.

4. Minor Recommendations

1. Correct the Use of “We” and “Our”: Align the writing style with the single-author perspective, indirect form.
2. Proofread the Manuscript: Ensure consistent formatting, proper punctuation, and grammar.
3. Enhance Clarity of Formulas: Provide a brief explanation after each major formula to improve readability for a broader audience.
4. Provide References for Stirling’s Formula: Include proper citations for foundational tools like Stirling’s approximation.

Reviewer Recommendation

While the manuscript addresses an important mathematical topic, the lack of clear purpose, minimal discussion, and unconventional organization significantly hinder its scientific impact. Substantial revisions are needed to improve clarity, structure, and contextual relevance before the manuscript can be considered for publication. My recommendation is major revision.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

There is no doubt that the paper represents the autor's own scientific result so I do not have any ethical concerns about the matter. Unfortunatly the paper style is far off from the accepted in the journal. For instance, it is inappropriate to write formulas in the abstract. The introduction section should contain a prehistory of the problem supplied with names and dates and only then some reasonings as a prerequisite before  the main results.  I mentioned these in my previous review report  but the remarks were not taken into account. Now, having  been impressed by the single authorship, I cannot incline the paper but decide that the major review is still required. As an example of the style,  I send you a published paper, you should follow it and I think your paper may be ready for publishing.

Author Response

Reviewer1: For instance, it is inappropriate to write formulas in the abstract.

My Comment: The text including formulas has been omitted from Abstract.

Reviewer1: The introduction section should contain a prehistory of the problem supplied with names and dates and only then some reasonings as a prerequisite before the main results.

My Comment: In the introduction, I wrote that ''There are only a few articles on approximating the Pochhammer product. One of them is \cite{Lampret-Poch}, where are given several approximations to the products in question. In our paper, we would like to present sharper and more general results than those given in \cite{Lampret-Poch}.'' Therefore, I emphasized that the prehistory of my article is quite modest.

Reviewer 2 Report

Comments and Suggestions for Authors

I am satisfied with the revision.

Author Response

Thank you for reviewing our manuscript.

Reviewer 3 Report

Comments and Suggestions for Authors

The reviewer now consider the manuscript appropriate to be published. 

Author Response

Thank you for reviewing our manuscript.

Round 3

Reviewer 1 Report

Comments and Suggestions for Authors

Neither all remarks have been taken into account nor them have been considered comprehensively.